Compositional Game Theory, compositionally

Atkey, Robert and Gavranović, Bruno and Ghani, Neil and Kupke, Clemens and Ledent, Jérémy and Nordvall Forsberg, Fredrik (2020) Compositional Game Theory, compositionally. In: Applied Category Theory 2020, 2020-07-06 - 2020-07-10.

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    We present a new compositional approach to compositional game theory (CGT) based upon Arrows, a concept originally from functional programming, closely related to Tambara modules, and operators to build new Arrows from old. We model equilibria as a module over an Arrow and define an operator to build a new Arrow from such a module over an existing Arrow. We also model strategies as graded Arrows and define an operator which builds a new Arrow by taking the colimit of a graded Arrow. A final operator builds a graded Arrow from a graded bimodule. We use this compositional approach to CGT to show how known and previously unknown variants of open games can be proven to form symmetric monoidal categories.